Diagonals and numerical ranges of direct sums of matrices

For any n-by-n matrix A, we consider the maximum number k=k(A) for which there is a k-by-k compression of A with all its diagonal entries in the boundary ∂W(A) of the numerical range W(A) of A. If A is a normal or a quadratic matrix, then the exact value of k(A) can be computed. For a matrix A of the form B⊕C, we show that k(A)=2 if and only if the numerical range of one summand, say, B is contained in the interior of the numerical range of the other summand C and k(C)=2. For an irreducible matrix A, we can determine exactly when the value of k(A) equals the size of A. These are then applied to determine k(A) for a reducible matrix A of size 4 in terms of the shape of W(A).